On the gonality of graph curves
type: Item
instance of: scholarly article
title: On the gonality of graph curves
zbMATH Open document ID: 1227.14034
author: E. Ballico
published in: Manuscripta Mathematica
publication date: 2009-08-27T00:00:00Z
review text: A graph curve \(X\) of genus \(g>2\) is a nodal curve, with \(2g-2\) smooth and rational, irreducible components, such that each of them intersects exactly three other components. These curves are stable, and they appear in the boundary of moduli spaces of smooth curves. Thus, the behaviour of linear series on graph curves is relevant for our knowledge of the Brill-Noether theory of general curves. The author studies the gonality of these reducible curves. As usual, when linear series on reducible curves are considered, the first task is to provide good definitions, to work with. The author defines the gonality of \(X\) as the minimum of the set \(\{d:\) there exists a strongly positive, spanned sheaf \(F\) of rank \(1\) and depth \(1\) on any component of \(X\), such that \(\deg(F)=d\}\). Then, the author shows that the gonality of a graph curve is \(2g-2\). It is interesting to notice that the gonality is computed by a sheaf \(F\) whose restriction to every component has degree \(1\). The sheaf \(F\) provides a map \(X\to\mathbb P^n\) which sends every component to a line.
reviewed by: Luca Chiantini
Mathematics Subject Classification ID: 14H51
Mathematics Subject Classification ID: 14H20
zbMATH DE Number: 5598924
zbMATH Keywords: stable curves
MaRDI profile type: Publication
full work available at URL: s00229-009-0253-8
OpenAlex ID: W2105771012
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DOI: 10.1007/S00229-009-0253-8
description: scientific article; zbMATH DE number 5598924